The Riesz representation theorem is a cornerstone of functional analysis, bridging linear functionals and Hilbert space elements. It states that every bounded linear functional on a Hilbert space can be uniquely represented as an inner product with a fixed vector.
This theorem has far-reaching implications in spectral theory, quantum mechanics, and operator analysis. It provides a concrete way to study abstract linear functionals, enabling deeper insights into the structure of Hilbert spaces and their applications in various fields of mathematics and physics.
Statement of theorem
Riesz representation theorem establishes a correspondence between linear functionals and elements of a Hilbert space
Provides a powerful tool for analyzing linear functionals in Spectral Theory by representing them as inner products
Bridges the gap between abstract linear functionals and concrete vector representations in Hilbert spaces
Top images from around the web for Hilbert space formulation Dual space (linear algebra) - Knowino View original
Is this image relevant?
Dual space (linear algebra) - Knowino View original
Is this image relevant?
1 of 3
Top images from around the web for Hilbert space formulation Dual space (linear algebra) - Knowino View original
Is this image relevant?
Dual space (linear algebra) - Knowino View original
Is this image relevant?
1 of 3
States that every bounded linear functional L L L on a Hilbert space H H H can be represented uniquely as an inner product
Representation takes the form L ( x ) = ⟨ x , y ⟩ L(x) = \langle x, y \rangle L ( x ) = ⟨ x , y ⟩ for some fixed y ∈ H y \in H y ∈ H
Norm of the functional equals the norm of the representing vector: ∥ L ∥ = ∥ y ∥ \|L\| = \|y\| ∥ L ∥ = ∥ y ∥
Applies to both real and complex Hilbert spaces
Extends the theorem to certain Banach spaces (reflexive spaces)
Identifies the dual space of a reflexive Banach space with the original space
Uses the notion of conjugate linear functionals instead of inner products
Requires additional conditions on the Banach space (uniformly convex)
Key concepts
Linear functionals, dual spaces, and inner products form the foundation of the Riesz representation theorem
Understanding these concepts aids in grasping the theorem's significance in Spectral Theory
Provides a framework for analyzing linear operators and their properties
Linear functionals
Defined as linear maps from a vector space to its underlying scalar field
Preserve vector addition and scalar multiplication
Can be represented as L : V → F L: V \rightarrow \mathbb{F} L : V → F where V V V is a vector space and F \mathbb{F} F is the scalar field
Play a crucial role in functional analysis and operator theory
Dual spaces
Consist of all continuous linear functionals on a given vector space
Denoted as V ∗ V^* V ∗ for a vector space V V V
Form a vector space themselves with pointwise addition and scalar multiplication
Provide insight into the structure and properties of the original space
Inner products
Generalize the notion of dot product to abstract vector spaces
Define a bilinear (or sesquilinear) map ⟨ ⋅ , ⋅ ⟩ : V × V → F \langle \cdot, \cdot \rangle: V \times V \rightarrow \mathbb{F} ⟨ ⋅ , ⋅ ⟩ : V × V → F
Satisfy properties of conjugate symmetry, linearity, and positive-definiteness
Enable the definition of orthogonality and norms in Hilbert spaces
Proof outline
Proof of the Riesz representation theorem involves three main steps
Demonstrates the power of Hilbert space geometry in functional analysis
Utilizes key properties of Hilbert spaces (completeness , orthogonality)
Existence of representation
Construct a closed subspace M = { x ∈ H : L ( x ) = 0 } M = \{x \in H : L(x) = 0\} M = { x ∈ H : L ( x ) = 0 }
Use the orthogonal decomposition theorem to write H = M ⊕ M ⊥ H = M \oplus M^\perp H = M ⊕ M ⊥
Show that M ⊥ M^\perp M ⊥ is one-dimensional and spanned by a vector y y y
Prove that L ( x ) = ⟨ x , c y ⟩ L(x) = \langle x, cy \rangle L ( x ) = ⟨ x , cy ⟩ for some scalar c c c
Uniqueness of representation
Assume two representations L ( x ) = ⟨ x , y 1 ⟩ = ⟨ x , y 2 ⟩ L(x) = \langle x, y_1 \rangle = \langle x, y_2 \rangle L ( x ) = ⟨ x , y 1 ⟩ = ⟨ x , y 2 ⟩
Show that ⟨ x , y 1 − y 2 ⟩ = 0 \langle x, y_1 - y_2 \rangle = 0 ⟨ x , y 1 − y 2 ⟩ = 0 for all x ∈ H x \in H x ∈ H
Conclude y 1 = y 2 y_1 = y_2 y 1 = y 2 using the positive-definiteness of inner products
Norm preservation
Prove ∥ L ∥ = sup ∥ x ∥ ≤ 1 ∣ L ( x ) ∣ = sup ∥ x ∥ ≤ 1 ∣ ⟨ x , y ⟩ ∣ \|L\| = \sup_{\|x\| \leq 1} |L(x)| = \sup_{\|x\| \leq 1} |\langle x, y \rangle| ∥ L ∥ = sup ∥ x ∥ ≤ 1 ∣ L ( x ) ∣ = sup ∥ x ∥ ≤ 1 ∣ ⟨ x , y ⟩ ∣
Use Cauchy-Schwarz inequality to show ∣ ⟨ x , y ⟩ ∣ ≤ ∥ x ∥ ∥ y ∥ |\langle x, y \rangle| \leq \|x\| \|y\| ∣ ⟨ x , y ⟩ ∣ ≤ ∥ x ∥∥ y ∥
Demonstrate that equality holds when x x x is a scalar multiple of y y y
Applications
Riesz representation theorem has wide-ranging applications in mathematics and physics
Provides a powerful tool for analyzing linear operators and functionals
Connects abstract functional analysis to concrete physical interpretations
Spectral theory
Enables the study of spectral properties of linear operators
Facilitates the representation of compact operators as infinite sums
Aids in the analysis of self-adjoint operators and their eigenvalue decompositions
Functional analysis
Provides a concrete realization of dual spaces in Hilbert spaces
Simplifies the study of weak topologies and weak convergence
Facilitates the development of operator theory and Banach algebra theory
Quantum mechanics
Allows representation of quantum observables as self-adjoint operators
Enables the formulation of the measurement postulate using projection operators
Facilitates the study of expectation values and uncertainty relations
Extensions and variants
Riesz representation theorem can be extended and modified for different settings
Adaptations allow for application to a broader range of mathematical spaces
Highlights the versatility and power of the theorem in various contexts
Complex vs real spaces
Theorem applies to both real and complex Hilbert spaces
Complex case requires conjugate linearity in the second argument of the inner product
Real case simplifies some aspects but may lose certain algebraic properties
Reflexive spaces
Extends the theorem to certain Banach spaces beyond Hilbert spaces
Requires the space to be reflexive (isomorphic to its double dual)
Provides a weaker form of representation using duality pairings instead of inner products
Weak topologies
Theorem plays a crucial role in understanding weak and weak* topologies
Enables characterization of weakly convergent sequences in Hilbert spaces
Facilitates the study of weak compactness and its applications
Historical context
Riesz representation theorem emerged during the development of functional analysis
Played a crucial role in formalizing the study of infinite-dimensional spaces
Influenced the direction of research in operator theory and spectral analysis
Development of functional analysis
Arose from the need to study integral and differential equations
Emerged in the early 20th century as a distinct field of mathematics
Integrated concepts from analysis, algebra, and geometry
Contributions of Frigyes Riesz
Hungarian mathematician who made significant contributions to functional analysis
Formulated the Riesz representation theorem in 1907
Also developed the spectral theorem for compact operators
Connections to other theorems
Riesz representation theorem relates to several other fundamental results in functional analysis
Forms part of a network of theorems that characterize linear functionals and operators
Provides a foundation for more advanced results in spectral theory
Hahn-Banach theorem
Extends bounded linear functionals from subspaces to the whole space
Complements Riesz representation by addressing existence of linear functionals
Applies to general normed spaces, not just Hilbert spaces
Lax-Milgram theorem
Generalizes Riesz representation to bounded bilinear forms
Provides conditions for existence and uniqueness of weak solutions to PDEs
Crucial in the study of variational problems and finite element methods
Spectral theorem
Characterizes normal operators on Hilbert spaces
Utilizes the Riesz representation to construct spectral measures
Fundamental in quantum mechanics and operator algebra theory
Examples and counterexamples
Illustrating the Riesz representation theorem with concrete examples aids understanding
Counterexamples highlight the limitations and necessary conditions for the theorem
Provides insight into the theorem's applicability in various mathematical settings
Finite-dimensional spaces
Theorem simplifies to the correspondence between vectors and linear functionals
Representation can be explicitly constructed using the standard basis
Dual space is isomorphic to the original space (R n ≅ ( R n ) ∗ \mathbb{R}^n \cong (\mathbb{R}^n)^* R n ≅ ( R n ) ∗ )
L^p spaces
Theorem applies directly to L 2 L^2 L 2 spaces, which are Hilbert spaces
For p ≠ 2 p \neq 2 p = 2 , L p L^p L p spaces are not Hilbert spaces but may be reflexive
Representation in L p L^p L p involves the Hölder conjugate q q q where 1 p + 1 q = 1 \frac{1}{p} + \frac{1}{q} = 1 p 1 + q 1 = 1
Non-reflexive spaces
Spaces like L 1 L^1 L 1 and L ∞ L^\infty L ∞ do not satisfy the conditions of the theorem
Dual of L 1 L^1 L 1 is L ∞ L^\infty L ∞ , but the converse does not hold
Illustrates the importance of reflexivity in extending the theorem to Banach spaces
Implications for operators
Riesz representation theorem has profound implications for the study of linear operators
Provides a framework for analyzing various classes of operators in Hilbert spaces
Facilitates the development of spectral theory and functional calculus
Adjoint operators
Theorem enables the definition of adjoint operators in Hilbert spaces
For an operator T T T , its adjoint T ∗ T^* T ∗ satisfies ⟨ T x , y ⟩ = ⟨ x , T ∗ y ⟩ \langle Tx, y \rangle = \langle x, T^*y \rangle ⟨ T x , y ⟩ = ⟨ x , T ∗ y ⟩
Crucial in the study of self-adjoint and normal operators
Compact operators
Representation theorem aids in characterizing compact operators
Allows decomposition of compact operators into sum of rank-one operators
Facilitates the study of spectral properties of compact operators
Self-adjoint operators
Theorem provides a powerful tool for analyzing self-adjoint operators
Enables spectral decomposition and functional calculus for self-adjoint operators
Fundamental in quantum mechanics for representing physical observables